Adaptive timestepping for conservation laws via adjoint error representation [Elektronische Ressource] / vorgelegt von Christina Steiner

Sujets

• Mathematics

Informations

• Publié par: rheinisch-westfalischen_technischen_hochschule_-rwth-_aachen
• Publié le: 01 janvier 2008
• Nombre de lectures: 5
• Langue: English
• Poids de l'ouvrage: 4 Mo

Extrait

Adaptive timestepping for conservation laws
via adjoint error representation
Von der Fakult at fur Mathematik, Informatik und
Naturwissenschaften der RWTH Aachen University zur
Erlangung des akademischen Grades einer Doktorin der
Naturwissenschaften genehmigte Dissertation
vorgelegt von
Diplom-Mathematikerin
Christina Steiner
aus Munc hen
Berichter: Univ.-Prof. Dr. Sebastian Noelle
apl. Prof. Dr. Siegfried Muller
Tag der mundlic hen Prufung: 19.12.2008
Diese Dissertation ist auf den Internetseiten der Hochschulbibliothek online
verfugbar.Acknowledgments
This thesis has been supported by the Deutsche Forschungsgemeinschaft in
the Collaborative Research Center SFB 401 "Flow Modulation and Fluid
Structure Interaction at Airplane Wings" of the RWTH Aachen.
First of all, I want to express my deepest gratitude to my supervisor
Prof. Dr. Sebastian Noelle. I am grateful to him for his steady support of
my studies, many fruitful discussions and for giving me the opportunity to
work on this project.
I also thank Prof. Dr. Siegfried Muller for many helpful discussions and
the received kind assistance on the work within the programming project
and taking on the task of being the second referee of this thesis.
I am also thankful for many interesting and fruitful discussions within
our Arbeitsgruppe, Andreas Bollermann, Roland Sch afer and Thies Frings,
for not only being colleagues but friends.
Special thanks go to all members of the Institut fur Geometrie und Prak-
tische Mathematik of RWTH Aachen University for their support and the
enjoyable and productive work environment
Last but not least, I want to thank Normann Pankratz for his multifar-
ious support, for keeping me on the right track, encouraging, and loving
me.
3Abstract
We study a recent timestep adaptation technique for hyperbolic conserva-
tion laws. The core of the method is a space-time splitting of adjoint error
representations for target functionals due to Suli [47] and Hartmann [29]. It
provides an e cient choice of timesteps for implicit computations of weakly
instationary ows. The timestep will be very large in regions of stationary
ow, and become small when a perturbation enters the ow eld. Besides
using adjoint techniques which are already well-established, we also add
a new ingredient which simpli es the computation of the dual problem.
Due to Galerkin orthogonality, the dual solution’ does not enter the error
representation as such. Instead, the relevant term is the di erence of the
dual solution and its projection to the nite element space, i.e. ’ ’ .h
We can show that it is therefore sucient to compute the spatial gradient
of the dual solution, w =r’. This gradient satis es a conservation law
instead of a transport equation, and it can therefore be computed with the
same algorithm as the forward problem, and in the same nite element
space. For this new conservative approach we will derive boundary condi-
tions. First we demonstrate the capabilities of the approach for a weakly
instationary test problem for scalar, 1D conservation laws. Then we extend
the computations to the 2D Euler equations, where we couple the adap-
tive time-stepping with spatial adaptation. For the spatial adaptation, we
use a multiscale-based strategy developed by Mul ler [38], which we com-
bine with the time adaptive method. The combined space-time adaptive
method provides an e cient choice of timesteps for implicit computations
of weakly instationary ows. The timestep will be very large in regions
of stationary ow, and becomes small when a perturbation enters the ow
eld. The e ciency of the solver is investigated by means of an unsteady
inviscid 2D ow over a bump.
56Contents
1 Introduction 11
1.1 Previous work . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Adjoint timestep control . . . . . . . . . . . . . . . . . . . . 13
1.3 Burgers’ equation on uniform grids . . . . . . . . . . . . . . 14
1.4 2D gas dynamics on adaptive spatial grids . . . . . . . . . . 14
1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . 15
2 Finite volume and Discontinuous Galerkin approximation 17
2.1 Model problem . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Discontinuous Galerkin and nite volume methods . . . . . 18
2.2.1 Finite volume methods . . . . . . . . . . . . . . . . . 22
3 A-posteriori error representation 25
3.1 The dual problem . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 Well-posedness of the dual problem . . . . . . . . . . 29
3.1.2 Examples of dual problems and target functionals . 29
3.2 A conservative form of the dual problem . . . . . . . . . . . 31
3.2.1 Rotation-invariance of the dual solution in two space
dimensions . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Space-time splitting and the error representation . . . . . . 38
3.4 Boundary conditions and functionals at the boundary . . . 41
3.4.1 A simplifying assumption . . . . . . . . . . . . . . . 41
3.4.2 Boundary conditions for the forward and the dual
problem . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4.3 Boundary condition for the gradient of the dual prob-
lem . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4.4 Example: functionals at the boundary . . . . . . . . 46
4 The space-time adaptive method 49
4.1 Numerical realization . . . . . . . . . . . . . . . . . . . . . . 51
4.1.1 Multiscale analysis - adaptation in space . . . . . . . 51
7Contents
4.1.2 Grid generation. . . . . . . . . . . . . . . . . . . . . 53
4.1.3 Newton method for the nonlinear system. . . . . . . 53
4.1.4 Computation of the dual problem . . . . . . . . . . . 54
5 Numerical examples: Burgers’ equation 55
5.1 Asymptotic decay rates . . . . . . . . . . . . . . . . . . . . 56
5.2 Computational results . . . . . . . . . . . . . . . . . . . . . 61
6 Numerical examples: 2D Euler equations 69
6.1 Setup of the numerical experiment . . . . . . . . . . . . . . 70
6.1.1 Test problem . . . . . . . . . . . . . . . . . . . . . . 70
6.2 Computational results . . . . . . . . . . . . . . . . . . . . . 78
6.2.1 Numerical strategies . . . . . . . . . . . . . . . . . . 78
6.2.2 Adaptive timesteps via adjoint indicator . . . . . . . 79
6.2.3e via ad hoc . . . . . . . 81
6.2.4 Uniform timesteps . . . . . . . . . . . . . . . . . . . 82
6.2.5 Newton iterations and linear iterations . . . . . . . . 86
6.2.6 Concluding remarks on the computational results to
2D Euler . . . . . . . . . . . . . . . . . . . . . . . . 89
7 Future work: Fluid structure interaction 91
7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2 The piston problem . . . . . . . . . . . . . . . . . . . . . . . 92
7.3 The dual and error representation for the piston
problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
8 Conclusion 97
A DG in time: implicit and explicit time discretisations 101
B 1D Euler equations 105
C 2D Euler equations 107
Nomenclature 109
List of Figures 111
8Contents
List of Tables 115
Bibliography 117
9Contents
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